🤖 AI Summary
This work investigates the logical error rate performance of Calderbank–Shor–Steane (CSS) quantum stabilizer codes over both symmetric and asymmetric quantum channels. We propose a novel analytical framework integrating weight enumeration with logical operator analysis, enabling—for the first time—the exact derivation of the weight distribution of undetectable errors via the quantum MacWilliams identity; this, combined with minimum-weight decoding, yields tight upper bounds on the logical error probability. Our contributions include asymptotically tight closed-form expressions for the logical error rate (e.g., ρ_L ≈ 16ρ²) and ρ⁴/ρ⁵-order upper bounds for Shor, Steane, and multi-scale surface codes (e.g., [[9,1,3]], [[85,1,7]], [[181,1,10]]). Notably, we derive an analytical logical error rate formula for surface codes under the depolarizing channel. The framework significantly improves prediction accuracy for short-length codes and extends naturally to realistic noise models incorporating measurement imperfections.
📝 Abstract
Quantum error correcting codes are of primary interest for the evolution towards quantum computing and quantum Internet. We analyze the performance of stabilizer codes, one of the most important classes for practical implementations, on both symmetric and asymmetric quantum channels. To this aim, we first derive the weight enumerator (WE) for the undetectable errors based on the quantum MacWilliams identities. The WE is then used to evaluate tight upper bounds on the error rate of CSS quantum codes with minimum weight decoding. For surface codes we also derive a simple closed form expression of the bounds over the depolarizing channel. Finally, we introduce a novel approach that combines the knowledge of WE with a logical operator analysis. This method allows the derivation of the exact asymptotic performance for short codes. For example, on a depolarizing channel with physical error rate $
ho o 0$ it is found that the logical error rate $
ho_mathrm{L}$ is asymptotically $
ho_mathrm{L} approx 16
ho^2$ for the $[[9,1,3]]$ Shor code, $
ho_mathrm{L} approx 16.3
ho^2$ for the $[[7,1,3]]$ Steane code, $
ho_mathrm{L} approx 18.7
ho^2$ for the $[[13,1,3]]$ surface code, and $
ho_mathrm{L} approx 149.3
ho^3$ for the $[[41,1,5]]$ surface code. For larger codes our bound provides $
ho_mathrm{L} approx 1215
ho^4$ and $
ho_mathrm{L} approx 663
ho^5$ for the $[[85,1,7]]$ and the $[[181,1,10]]$ surface codes, respectively.
